signal analysis and processing in biomedicine analysis and processing in...overview o mathematical...

Signal Analysis and Processing in Biomedicine

Prof. dr. Srdjan Stankovic

Prof. dr. Irena Orovic

LECTURE NOTES

Development and development of this material is partly supported by Project

Studies in Bioengineering and Medical Informatics (BioEMIS)

For subject: Biomedical signal and image processing

Izrada i štampanje ovog nastavnog materijala je djelimicno pomognuta od

strane Projekta

Studije u bioinženjeringu i medicinskoj informatici

Za predmet: Digitalna obrada biomedicinskih signala i slika

TEMPUS-1-2012-1-UK-TEMPUS-JPCR

The authors are very thankful for support

Podgorica – January - 2016

TEMPUS BioEMIS Edition, 2016 2

Overview

O Mathematical transforms in biomedical signal

processing

O Computer tomography

O Compressive sensing

O Compressive sensing in biomedical imaging

O ECG signals

O Hermite transform in ECG signals analysis

O Detection of swallowing sounds – appl.

Dysphagia

O Telemedicine


Mathematical transforms in biomedical signal

processing


Fourier transform

time frequency

( ) ( ) j tF f t e dt

1( ) ( )

2

j tf t F e d

inverse:

• Some useful Properties:

( ) ( ) ( ) ( ) ( ) ( )j t j t j tf t g t e dt f t e dt g t e dt F G

Linearity

00( ) ( )

j tj tf t t e dt e F

Time shift

00( ) ( )

j t j te f t e dt F

Frequency shift

*{ ( ) ( )} ( ) ( ) ( ) ( )FT f t g t FT f g t d F G

Convolution

*{ ( ) ( )} ( ) ( )FT f t g t F G TEMPUS BioEMIS Edition, 2016 5

Discrete Fourier transform

0 50 1000

20

40

60

0 50 100-3

-2

-1

0

1

2

Noisy signal

1 2 ...

N-1

f(n)

0

21

0

( ) ( )N j nk

N

n

DFT k f n e

21

0

1( ) ( )

N j nkN

k

f n DFT k eN

time frequency TEMPUS BioEMIS Edition, 2016 6

Fourier transform

( )( ) j tx t Ae -Frequency modulated signal

Ideal time-frequency representation of

x(t) should concentrate energy along

the instantaneous frequency of the

signal. It is defined as:

2( , ) 2 ( - '( )),ITF t A t

ω=’(t)

is instantaneous frequency

where


( , ) ( ) ( ) jSTFT t w x t e d

• Short Time-Fourier Transform

window function

1

( , ) ( , )m

M

xm

STFT t STFT t

STFT is linear

transform 1

( ) ( )M

mm

x t x t

Time-Frequency Representation

Multicomponent signals:


S-method • S-method: Distribution with the auto-terms being the same as in the

Wigner distribution, but with reduced or completely removed cross-

terms: 1( , ) ( ) ( , ) *( , ) ,SM t P STFT t STFT t d

*

2 *

1

( , ) ( ) ( , ) ( , )

( , ) 2 Re ( , ) ( , )

d

d

d

L

i L

L

i

SM k n P i STFT k i n STFT k i n

STFT k n STFT k i n STFT k i n

Where 𝑃(𝜃 is finite frequency domain window. Discrete form of the S-

method is:

In order to avoid the presence of cross-terms, the

value of Ld should be less than half of the distance

between two auto-terms


Wavelet transform Continuous wavelet transform:

O Wavelets can be used as basis for representing functions;

O Wavelets-functions formed by scaling and translation of basis (mother) functions in the time domain;

O It satisfies the following conditions:

Area under the curve is equal

to zero

The function is square

integrable

( ) 0t dt

2( )t dt

( ) ( )i i

i

y t a t


Wavelet transform

O Continuous wavelet transform of the signal is given by:

,( , ) ( ) ( )a bW a b t f t dt

,

1( ) ( ) ( , )a b

a b

f t t W a b dadbC

Inverse form:

2( )

C d

-Fourier transform of ( )t𝛹(𝜔

C is positive and finite

• Wavelet is defined by

formula:

,

1( )a b

t bt

aa

- a, b - two real numbers used for

scaling and translation in time

basis function is shrunk in time

basis function is spread

0 1a

1a


Wavelet families

O Daubeshies

O Haar

O Biorthogonal

O Mexican Hat

O Symlets

O Morlet

O Coiflets

O Meyer

Haar

Daubechies Morlet Symlet

Biorthogonal wavelet

function


Discrete wavelets

0ma a 0 0

mb nb a• Discretization is done

by using:

,

m, n Z, b0>0

/ 2, 0 0 0( ) ( )m m

m n t a a t nb Discrete wavelet

function

/ 2, 0 0 0( ) ( )d m m

m nW a f t a t nb dt

Discrete wavelet

transformation

• Using a0=2 and b0=1 dyadic sampling is obtained

/ 2, ( ) 2 (2 )m m

m n t t n / 2

, 2 ( ) (2 )d m mm nW f t t n dt


O Decomposition is applied to rows, then to columns

Standard wavelet decomposition

Li - low-frequency sub-band

Hi - high-frequency sub-band


Quincunx decomposition O Decomposition uses only low frequency sub-band of different level

O Low frequency sub-band is then divided into low and high frequency part

ORIGINAL

IMAGEL1 H1

L2

H1

H2

H1

H2

H1

H2

H3

L4

H4


Pyramidal wavelet decomposition

Uniform wavelet decomposition

LLLL LLHL HLLL HLHL

LLLH LLHH HLLH HLHH

LHLL

LHLH

LHHL

LHHH

HHLL HHHL

HHLH HHHH


Hermite functions

2 2/2 /20 14 4

1 2

1 2( ) , ( ) ,

12( ) ( ) ( ), 2.

x x

p p p

xx e x e

px x x x p

p p

Hermite functions of

different orders

Recursive realization of

Hermite functions:

The Hermite functions provides good

localization and the compact support

in both time and frequency domain.

The i-th order Hermite function is

defined as:

2 2/2( 1) ( )( ) .

2 !

i t i t

i ii

e d et

dti


Hermite expansion ( ) (1)

( ) (1) ,X P X

x i X iP

1,...,i P

( ) ( ) ( )f i X i x i

First step –removing the

baseline

Baseline substraction from the original

signal 1

0( ) ( )

N

p pp

f i c i

Decomposition using N Hermite functions

( ) ( ) ( )p pc i f i i di

Hermite expansion cofficients:

11

1( ) ( ) ( )

M pp m mM

mc i x f x

M

Gauss-Hermite quadrature technique can

be used to calculate the Hermite expansion

coefficients

mx

- a zeros of an M-th order Hermite

polynomial

1 21

( )( )

( ( ))

p mpmM

M m

xx

x


Hermite transform

O The expansion using N Hermite functions can be written

in matrix form. First, we define the Hermite transform

matrix W (of size ):

0 0 0

2 2 2

1 1 1

1 1 1

2 2 2

1 1 1

1 1 1

2 2 2

1 1 1

(1) (2) ( )

( (1)) ( (2)) ( ( ))

(1) (2) ( )1

( (1)) ( (2)) ( ( ))

(1) (2) ( )

( (1)) ( (2)) ( ( ))

N N N

N N N

N N N

N N N

M

M

M

MN

M

M

W

If the vector of Hermite

coefficients is: 0 1 1, ,...,T

Nc c c c

and vector of M signal samples is:

[ (1), (2),..., ( )]Tf f f Mf

then we have

c WfFor a signal of length M, the complete set

of discrete Hermite functions consists of

exactly N=M function. In some

applications, a smaller number of Hermite

functions N<M can be used TEMPUS BioEMIS Edition, 2016 20

O Having in mind the Gauss-Hermite approximation, the

inverse matrix contains N Hermite functions, given

by:

O Now, the Hermite expansion for the case of discrete signal

can be defined as follows:

1W

Hermite transform

0 0 0

1 1 1 1

1 1 1

(1) (2) ( )

(1) (2) ( )

(1) (2) ( )N N N

M

M

M

Ψ W

1 f W c Ψc


Computer tomography


Computer Tomography O Clinical Computed Tomography (CT) was introduced

in 1971 - limited to axial imaging of the brain in neuroradiology

O It developed into a versatile 3D whole body imaging modality for a wide range of applications in for example O oncology,

O vascular radiology,

O cardiology,

O traumatology and i

O interventional radiology.

O Computed tomography can be used for diagnosis and follow-up studies of patients, planning of radiotherapy treatment, screening of healthy subpopulations with specific risk factors. TEMPUS BioEMIS Edition, 2016 23

O CT scanning is suitable for 3D imaging of the brain,

cardiac, musculoskeletal, and whole body CT imaging.

O The images can be viewed as colored 3D rendered

images, although radiologists prefer black and white 2D

images

Computer Tomography


O Acquisition is achieved using:

O hundreds of detector elements along the detector arc (800-900 detector elements),

O using rotation of the x ray tube around the patient, taking about 1000 angular measurements

O using tens or even hundreds of detector rows aligned next to each other along the axis of rotation

Computer Tomography


Image reconstructions from projections

O Image reconstruction based on projections has

important applications in various fields (e.g., in

medicine when dealing with computer

tomography, which is widely used in

everyday diagnosis).

O Consider an object in space, which can be

described by the function f(x,y). The projection

of function f(x,y) along an arbitrary direction

(defined by an angle ) can be defined as

follows:

( ) ( , ) , cos sinAB

p u f x y dl u x y TEMPUS BioEMIS Edition, 2016 26


The previous equation can be written as follows:

( ) ( , ) ( cos sin )p u f x y x y u dxdy

The Fourier transform of the projection is given by:

( ) ( ) j uP p u e du

Furthermore, the two-dimensional Fourier transform of f(x,y) is defined as:

( )( , ) ( , ) x yj x y

x yF f x y e dxdy



As a special case, we can observe ( , )x yF for ωy=0

0 0

( ,0) ( , )

( ) ( )

x

x

j xx

j x

F f x y e dxdy

p x e dx P

To conclude: the Fourier transform of a

two-dimensional object along the axis

ωy=0 is equal to the Fourier transform

along the projection angle =0

In the rotated coordinate system, we have:

cos sin

sin cos

u x

l y

( cos sin )

( ) ( ) ( , )

( , ) ( , ),

j u j u

j x y

P p u e du f u l e dudl

f x y e dxdy F

cossin

( , ) ( , )x

y

x yF F

where TEMPUS BioEMIS Edition, 2016 28

O Therefore, we can conclude:

O If we have the object projections, then we can determine their Fourier transforms.

O The Fourier transform of a projection represents the transform coefficients along the projection line of the object.

O By varying the projection angle from 0 to 180 we obtain the Fourier transform along all the lines (e.g., we get the Fourier transform of the entire object), but in the polar coordinate system.

O To use the well-known FFT algorithm, we have to switch from polar to rectangular coordinate system. Then, the image of the object is obtained by calculating the inverse Fourier transform.

O The transformation from the polar to the rectangular coordinate system can be done by using the nearest neighbor principle, or by using some other more accurate algorithms that are based on the interpolations.



Compressive sensing


Compressive sensing O Compressive sensing approaches have been

intensively developed to overcome the limits of traditional sampling theory by applying a concept of compression during the sensing procedure.

O Compressive sensing aims to provide the possibility to acquire much smaller amount of data, but still achieving the same quality (or almost the same) of the final representation as if the physical phenomenon is sensed according to the conventional sampling theory.

O In that sense, significant efforts have been done toward the development of methods that would allow to sample data in the compressed form using much lower number of samples.


Compressive sensing O Compressive sensing opens the possibility to simplify

very expensive devices and apparatus for data recording,

imaging, sensing (for instance MRI scanners, PET

scanners for computed tomography, high resolution

cameras, etc.).

O Furthermore, the data acquisition time can be

significantly reduced, and in some applications even to

almost 10 or 20% of the current needs.


WE WILL NEED 6 TIMES LESS MEASUREMENTS

FOR THE SAME IMAGE QUALITY

IT MEANS 6

TIMES LOWER

EXPOSURE OF

PATIENTS

TO INVASIVE

METHODS TEMPUS BioEMIS Edition, 2016 34

Compressive sensing

O If the samples acquisition process is linear, than

the problem of data reconstruction from acquired

measurements can be done by solving a linear

system of equations.

O Measurement process can be modelled by the

measurement matrix .

O Signal f with N samples can be represented as a

signal reconstruction problem using a set of M

measurements obtained by using as follows:

O where y represents the acquired measurements.

Φf = y


CS conditions O CS relies on the following conditions:

Sparsity – related to the signal nature;

O Signal needs to have concise representation when

expressed in a proper basis (K<<N)

Incoherence – related to the sensing modality; It

should provide a linearly independent

measurements (matrix rows)

Random undersampling is crucial

Restriced Isometry Property – is important for

preserving signal isometry by selecting an

appropriate transformation TEMPUS BioEMIS Edition, 2016 36

Standard approach for signal sampling and

its compressive sensing alternative


CS problem formulation

O The method of solving the undetermined system of

equations , by searching for the

sparsest solution can be described as:

y=ΦΨx Ax

0min subject to x y Ax

0x l0 - norm

• We need to search over all possible sparse vectors x with

K entries, where the subset of K-positions of entries are

from the set {1,…,N}. The total number of possible K-

position subsets is N

K


O A more efficient approach uses the near optimal solution

based on the l1-norm, defined as:

1min subject to x y Ax

CS problem formulation

• In real applications, we deal with noisy signals.

• Thus, the previous relation should be modified to include

the influence of noise:

1 2min subject to x y Ax

2e

L2-norm cannot be used because the minimization problem

solution in this case is reduced to minimum energy solution,

which means that all missing samples are zeros TEMPUS BioEMIS Edition, 2016 39

Signal - linear combination of the

orthonormal basis vectors

Summary of CS problem formulation

1

( ) ( ), : .N

i ii

f t x t or

f =Ψx𝐟

y=ΦfSet of random measurements:

random measurement

matrix

transform

matrix

transform

domain vector


CS reconstruction algorithms O I group: convex optimizations such as

O Basis pursuit

O Dantzig selector,

O Greedy algorithms:

O Matching pursuit,

O Orthogonal matching pursuit;

O Iterative thresholding (hard and soft versions)

O Hybrid methods:

O Compressive sampling matching pursuit


f

Transform matrix

Measurement matrix

y Measurement vector

Greedy algorithms – Orthogonal Matching Pursuit

(OMP)


Iterative hard thresholding O The IHT algorithm is an iterative method

where Hk is the hard thresholding operator that sets all but the k largest (in magnitude) elements in a vector to zero.

O IHT is a simple reconstruction algorithm and it can recover sparse and approximately sparse vectors with near optimal accuracy.

O 1) The step size µ has to be chosen appropriately to avoid instability of the method

O 2) IHT has only a linear rate of convergence


y – measurements

M - number of measurements

N – signal length

T - Threshold

• DFT domain is assumed as sparsity domain

• Apply threshold to initial DFT components

(determine the frequency support)

• Perform reconstruction using identified support

Single-Iteration Reconstruction

Threshold based Algorithm for components

100 200 300 400 500

0

0.5

1

1.5

2

2.5

3

3.5

Initial FT and Threshold

100 200 300 400 500

0

0.5

1

1.5

2

2.5

3

3.5

Reconstructed FT

100 200 300 400 500

0

0.5

1

1.5

2

2.5

3

3.5

Initial FT

100 200 300 400 500

0

0.5

1

1.5

2

2.5

3

3.5

Reconstructed FT


Variance of DFT values

at the non-signal and

signal positions can be

calculated as:

2

1

( )var{ }

( 1)i

K

k k i

k

M N MF A

N

y=ΦΨx=Θx Is a CS matrix (could be DFT,

DCT, HT) TEMPUS BioEMIS Edition, 2016 45

In each iteration we

need to remove the

influence of

previously detected

components and to

update the value of

threshold

Iterative threshold based solution for detection of

desired components


Compressive sensing in biomedical imaging


Compressed sensing in CT

O Due to its powerful ability of reconstructing signal or image from highly undersampled data, CS has attracted tremendous attention in medical imaging community.

O The first attempt to apply CS to medical imaging was done by Lustig et al., in which they successfully reconstructed MR images with 40% of the data.

O There are also research efforts on compressed sensing CT because in principle a CS based algorithm can produce good reconstructions using fewer samples (projections), it has the potential to reduce the radiation exposure to the patient.


Recovering of highly undersampled signal

512*512 Shepp-Logan Undersampled by 22 radial lines

Normal Reconstruction

CS reconstruction algorithm TEMPUS BioEMIS Edition, 2016 49

O The most popular type of method that has

been studied is the Total Variation (TV) based

compressed sensing methods.

O TV based methods use total variation as a

sparsity transform.

Compressed sensing in CT


Total Variation minimization O Natural images are not sparse – neither in space

nor in the frequency domain

O Instead of the l1 norm minimization, Total Variation

(TV) minimization is commonly used

O Having in mind that the image gradient is sparse,

TV is, in fact, l1 norm minimization of the image

gradient

O Introduced by Rudin, Osher and Fatemi, for

solving the inverse problems


O Inverse problems:

O signal X, measurements y (modeled by applying the operator A – measurement matrix)

O Noisy measurements: y=AX+n

O Linear inversion (if A is linear), deconvolution problem

O Standard approach to linear inversion problems:

O Definition of the objective function

O Solution of the minimization function according to X

Total Variation minimization


y nAX


X - signal to be estimated

n – aditive noise

A - matrix that models measurement

process Defining the objective

function:

( ) ( , ) ( )F d R yX AX X

– difference measure between signals y and X

- it can be mean square error

R(X) – regularization function

λ – regularization parameter, λ>0

The value of the X should be such that the relation AX

corresponds to the vector y

2

2( , )

ld y AX y AX

d(y, AX)


O Solution X=A-1y is not applicable if A is not an

invertible matrix

O The goal of R(X) is to avoid such situations

O If the signal X is sparse, the function R(X)

corresponds to the l1 norm, i.e. objective

function is defined as:


2 1

2( )

l lF X y AX X


O The regularization function R(X)

can be defined as TV norm:

O The is the gradient operator:

O TV norm in the discrete form:


1

( )l

R X X

,

( 1, ) ( , )

( , 1) ( , )i j

i j i j

i j i j

X XX

X X𝛻

2 2

,

( ) ( ) ( ) ,h v

i j

TV X XX( 1, ) ( , )

( , 1) ( , )

h

v

i j i j

i j i j

X

X

X X

X X

- Row and column differences TEMPUS BioEMIS Edition, 2016 55

O Special attention is paid to the Iterative shrinkage/thresholding (IST) algorithms

O Used for solving multi-dimensional optimization problems.

O Problem of these algorithms: slow convergence

O Convergence of the IST algorithms is speed up by introducing the two step IST algorithm (two-step Iterative shrinkage/thresholding, TwIST)

Iterative shrinkage/thresholding algorithms


Two-step Iterative shrinkage/thresholding, TwIST

O Finds solution X in two-steps:

O Where is defined as:

O Denoising and regularization functions are described by

following relations:

1 0

1 1

( ),

(1 ) ( ) ( ), 1.t t t t t

X X

X X X X

( ) ( ( ))T X X A y AX

21 1( ) arg min ( )

2reg

X

X X

( )reg iiD X X

Denoising operator

Regularization function for TV l1 problems TEMPUS BioEMIS Edition, 2016 57

O Optimal choice for the parameters α and β:

O There is a large number of methods for solving TV minimization problems: time marching scheme, fixed point iteration metod, majorization-minimization approach

O TV-L2 problem can be defined as:

Two-step Iterative shrinkage/thresholding, TwIST

2

1

1 21,

max( )1 m

k

k

1

1

0 ( ) ,

max( )

T

i m

m

k

A A

2 2min

2ii

D

X

X AX y

ɛ - constant

y measurements

X –image to be obtained

A-CS matrix TEMPUS BioEMIS Edition, 2016 58

Image reconstruction examples

Original

Estimate

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

Estimate

50 100 150 200 250

50

100

150

200

250

Estimate

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

Reconstructed images

DFT masks

Original image


Image reconstruction – more examples

Reconstructe

d image

Reconstructe

d image

Reconstructe

d image


ECG signals


ECG signals -basics

O ECG signal represents the activity of the heart.

O The propagation of electrical waves resulted from cardiac activity can be measured as a difference of potentials between two points on the human body

O In the case of multi-channel ECG, the signals are measured between different pairs of electrodes on the human body.

O ECG signal characteristics can be described using 7 specific parts.

O P,Q,R,S and T peaks

O 2 iso-electric parts that separate P and T peak from the large QRS complex.


ECG signals -basics


ECG-signal basics O The heart rhythm is of sinus type characterized among other features

by the presence of P-complex before every QRS complex.

O The heart impulse rate is characterized by heart agitation in average in the frequency of 60–100 beats per minute (bpm) for adults, although it might be normal, especially for athletes, to have resting bpm as low as 30.

O Deviations of the rate beyond this range can be treated as abnormal case. However, the limit frequencies should be taken individually.

O For the time duration of PR, QRS and QT parts in the cardiac cycle, a reasonable rule is to consider that the interval QT is less than a half of the distance between two successive QRT complexes. That is, QT should be less than 1/2 of the RR interval.

O P complex (P wave) is normally 0.04–0.11 s in duration. Its deviation from the normal wave shape or its disappearing means a pathological case.

O The normal duration of ST segment is 0.02–0.12 s. Any drop in the duration of the ST segment suggests ischemic, whilst its shift above the cycle-axis suggests a heart attack.

O The normal T-complex is about half of the P-complex time. When it is inverted (except for aVR lead), it typically indicates a heart attack.


Recorded evolution of ECG waveform showing the

pathological disturbances after the heart attack in ST, T and

Q parts


Finding of proper heartbeats on the example of

R and P points. The signal and the markings of

characteristic points are drawn by the simulation

software.


• The signal split into 0.25-mV high and 30-second wide spans

• Inside these spans all minima and maxima are counted. Based on their

numbers the span of the signal is estimated and the mV spans containing

possible R and S points are acquired.

• On the basis of these spans the probable P and T points are found.

• The last part is the search for Q point and ST segment with the use of least

squares approach.

• A training algorithm can be used to obtain the ECG signal pattern

The algorithm is based on statistical analysis of probabilities of the existence of

characteristic points of ECG signal in the given millivolt (mV) and time frames.


ECG signals and QRS complexes

O ECG signals represent the records of electrical activity of the heart over a period of time

O QRS complexes are the most characteristic waves in ECG signals

O QRS complexes are crucial in different stages of medical diagnosis and treatment of heart deceases

O Important research efforts have been made in the development of recognition, classification and compression algorithms for QRS complexes

O Effective storage, automatic detection of anomalies and automatic diagnosis based on the processing of QRS complexes are the state-of-the-art aims in modern biomedicine


QRS complexes in ECG signals

-0.06 -0.04 -0.02 0 0.02 0.04 0.06-0.8

-0.6

-0.4

-0.2

0

0.2

t [ms]

Am

plit

ude m

V

First 10 miliseconds of a real ECG

signal

Red parts denote intervals of QRS

complexes

PhysioNet: MIT-BIH ECG Compression Test

Database,

http://www.physionet.org/physiobank/database/cdb

Q

R

S

0 1 2 3 4 5 6 7 8 9 10-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

t [ms]

Am

plitu

de

mV


QRS complexes and the Hermite transform

O The p-th order Hermite function and the functions of the p-1 and p-2 orders and can be related with the following recursive formula:

O

𝜓0(𝑡, 𝜎 =1

𝜎 𝜋𝑒−

𝑡2

2𝜎2 , 𝜓1(𝑡, 𝜎 =2

𝜎 𝜋

𝑡

𝜎𝑒−

𝑡2

2𝜎2 ,

𝜓𝑝(𝑡, 𝜎 =𝑡

𝜎

2

𝑝 𝜓𝑝−1(𝑡, 𝜎 −

𝑝−1

𝑝 𝜓𝑝−2(𝑡, 𝜎 .

O Hermite tranform is defined by:

O 𝑓(𝑡 = 𝑐𝑝𝜓𝑝(𝑡, 𝜎 𝑀

𝑝=0, while the Hermite coefficients are:

O 𝑐𝑝 =1

𝑀

𝜓𝑝(𝑡𝑚,𝜎

𝜓𝑀−1(𝑡𝑚,𝜎 2 𝑓(𝑡𝑚 𝑀

𝑚=1, 𝑝 = 0,1, . . . , 𝑀 − 1


O Visual similarity between QRS

complexes as signals with

compact time support and

Hermite bass functions is obvious

O Thus, Hermite transform is often

incorporated in classification and

compression algorithms for QRS

complexes.

O A QRS complex can be modeled

as:

O s(𝑡 = 𝑐𝑝𝜓𝑝(𝑡, 𝜎 𝐾

𝑝=0

O Here, K is the number of non-zero

Hermite coefficients needed for

the successful representation of

QRS complex

0 10 20 30 40 500

0.5

1

0(tn)

0 10 20 30 40 50-1

0

1

1(tn)

0 10 20 30 40 50-1

0

1

2(tn)

0 10 20 30 40 50-1

0

1

tn

3(tn)

First four Hermite basis

functions

Only few Hermite coefficients are

needed for the representation of

QRS complexes


QRS complexes and the Hermite transform: Representation and the CS scenario


The influence of the Hermite scaling factor

O Hermite transform of the

QRS complex:

O first row – the selected QRS

complex,

O second row – Hermite

coefficients of the signal with

σ = 1 ⋅ 𝛥𝑡 ,

O third row – Hermite

coefficients of the signal with

σ = 5.7 ⋅ 𝛥𝑡 .

O Automatic procedure of

scaling factor optimization?

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1

-0.5

0

0.5

time [s]

Am

pli

tud

e [m

V]

0 5 10 15 20 25 30 35 40 450

0.5

1

p

|cp|

0 5 10 15 20 25 30 35 40 450

0.5

1

p

|cp|


A compression procedure O Use a suitably chosen scaling factor

O Aproximate the signal with K most significant HT

coefficients, such that the relative error:

𝐸 =‖𝐟 −𝐟‖2

‖𝐟‖2 < 10%

where 𝐟 is the inverse HT of the K most significant

coefficients, and f is the vector of the original signal

O Our experiments show that only 4-5 Hermite coeffcients

are needed to be stored in order to preserve a medically

acceptable error level.

O Very important in order to generate databases for every

patient with heart desease, in order to make efficient

diagnosis and anticipation of future problems TEMPUS BioEMIS Edition, 2016 74

A compression procedure O Original and approximated QRS complex

O Only 4 HT coefficients are used, signal optimally resampled

-0.05 0 0.05

-1

-0.5

0

0.5

1

(a)

Amplitude [mV]

nT [s]

Original QRS complex Hermite transform of the original QRS complex

0 5 10 15 20 25

-200

-100

0

100

200

(b)

p

-0.05 0 0.05

-1

-0.5

0

0.5

1Amplitude [mV]

(c)

Signal resampled by using optimal and m

tmT [s]

0 5 10 15 20 25-400

-300

-200

-100

0

Hermite transform of the resampled signal

p

(d)


Challenges

O Automatic optimization of the scaling factor using concentration measures

O Apply concentration measures in order to determine a threshold for the automatic recognition and/or classification of QRS complexes in ECG signals

O Use the Hermite transform as the core in expert systems and classification algorithms

O Results verification in consultation with cardiologists


Detection of swallowing sounds – appl. Dysphagia


Scope of the approach

O Innovative approach for the analysis of one-dimensional biomedical signals that combines the Hermite projection method with time-frequency analysis

O A two-step approach to characterize vibrations of various origins in swallowing accelerometry signals

O First, by using time-frequency analysis we obtain the energy distribution of signal frequency content in time.

O Second, by using fast Hermite projections we characterize whether the analyzed time-frequency regions are associated with swallowing or some other phenomena (vocalization, noise, bursts, etc.).

O The numerical analysis of the proposed scheme clearly shows that by using a few Hermite functions, vibrations of various origins are distinguishable.


1. SWALLOWING DIFFICULTIES

O Deglutition, or swallowing, is a well-defined, complex process of transporting food or liquid from the mouth to the stomach.

O Swallowing consists of four phases: oral preparatory, oral, pharyngeal, and esophageal.

O Patients suffering from dysphagia (swallowing difficulty), usually deviate from the well-defined pattern of healthy swallowing.

O Dysphagia is a common problem encountered in the rehabilitation of stroke patients, head injured patients, and others with paralyzing neurological diseases

O Today's dysphagia management relies heavily on the videofluoroscopic swallowing study (VFSS).

O VFSS is accepted as the gold standard, but requires expensive X-ray equipment and expertise from speech-language pathologists and radiologists.


2. Cervical auscultation and Swallowing accelerometry

O Cervical auscultation is a promising non-invasive tool for the

assessment of swallowing disorders, adopted by dysphagia

clinicians.

O Cervical auscultation approach involves the examination of

swallowing signals acquired via a stethoscope or other acoustic

and/or vibration sensors during deglutition.

O Swallowing accelerometry, a technique that involves an

accelerometer placed on the neck to monitor vibrations

associated with swallowing activities, has been used to detect

aspiration in several studies

O Nevertheless, the presence of various vibrations not associated

with swallowing can severely contaminate swallowing

accelerometry signals

O Vocalizations either voluntarily or involuntarily can have an

adverse effect on these signals, whose presence masks the

observed swallowing signals. TEMPUS BioEMIS Edition, 2016 80

3. Data O The sample data considered in this paper were

gathered over a three month period from a public science centre in Toronto, Ontario, Canada. A dual-axis accelerometer (ADXL322, Analog Devices) was attached to the participant’s neck using double-sided tape in order to monitor vibrations associated with swallowing.

O Data were band-pass filtered in hardware with a pass band of 0.1-3000 Hz and sampled at 10kHz using a custom LabVIEW program running on a laptop computer. Data were saved for subsequent off-line analysis


During data collection, participants were

cued to perform three types of swallows

involving saliva and water swallows. The

entire data collection session lasted 15

minutes per participant. The participants

were instructed not to vocalize.

Nonetheless, approximately one quarter of

all recordings contained either voluntary or

involuntary vocalizations.


Time-Frequency Analysis

O Signals considered in this paper mostly contain

swallowing vibrations.

O Some recordings also contain vibrations associated

with vocalization (e.g., speech, cough, laughter) and

various burst components produced by the equipment

and noise.

O The most dominant vibrations are those produced by

swallowing and vocalization.

O Fortunately, vibrations associated with different

phenomena provide unique time-frequency signatures.


O Thus, time-frequency representations are crucial for the analysis

and classification of these signals. Also, having in mind

multicomponent nature of these signals, time-frequency

representations without cross-terms should be used.

Time-Frequency Analysis

1500 2000 2500

150

200

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300

350

1000 1500 2000

200

250

300

350

Time-frequency regions of spectrogram

for voiced sound (left), for swallowing

sound (right) TEMPUS BioEMIS Edition, 2016 84

Hermite projection method applied to time-frequency regions

O To understand differences between swallowing vibrations and other miscellaneous vibrations (speech, laugh, cough, etc.) we observe the structures in the time-frequency regions. Here, the spectrogram, as the simplest time-frequency distribution, is used:

O A certain pre-processing in classification procedure is applied, since the signal could be corrupted by noise: the time-frequency mask is defined to remove the noise influence and locate the significant signal components:

O where the threshold value ξ, i.e., the energy floor is obtained as:

22

( , ) ( , ) ( ) ( ) jSPEC t STFT t x t w e d

1 for ( , )( , )

0

SPEC tL t

otherwise

log10(max( ( , ))),10

SPEC tt


O Filtered spectrogram is given by:

O The energy vector is calculated to determine the time intervals

containing vibration activities:

O Consequently, the time support vector is obtained as (threshold

β is used to remove the remaining noise):

O The time-frequency regions for classification are extracted from

for

O The fast Hermite projection method is applied to these regions


( , ) ( , ) ( , )filtSPEC t L t SPEC t

( ) ( , )filtE t SPEC t

1 for ( )( ) ,

0f

E tE t

otherwise

( , )filtSPEC t arg ( ) 1 .ft E t


500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500

50

100

150

200

250

300

2 1 3 4 5 6 7 8 9 10 11 12 13

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500

10

20

30

40

50

60Filtered time-frequency representation and assigned regions

(upper row), time support vector (bottom row) TEMPUS BioEMIS Edition, 2016 87

O We use a small number of functions for reconstruction such

that:

O simple structure regions are reconstructed with a small error

O complex structure regions with a significantly larger error.

O The difference between the original and reconstructed regions

i.e. the error, which depends on the region’s structure of a

region, will be used for characterization. This difference is

measured by the mean squared error, as follows:

O where R(t,) denotes the original region, R’(t,) is the

reconstructed region


21( ( , ) '( , ))

t

MSE R t R tT


O It has been shown that the voiced

vocalization regions (speech, cough,

laughter, etc.) are more complex,

and thus have the higher MSE, in

comparison comparing to the

regions with containing swallowing

sounds vibrations.

O Moreover, there is a significant gap

between the values of MSE for

voiced vocalization and for

swallowing sounds vibrations.

O On the other hand, the MSE for

regions with swallowing sounds

vibrations is much higher than for

the regions with containing noise.

O The difference between the original

and reconstructed regions, given in

terms of MSE, is used as a

parameter for characterizing to

characterize a regions

No. Region

description MSE

1 Noise 0.086

2 Noise 0.01

3 Noise 0.22

4 Swallowing

vibrations

36.29

5 Burst 0.13

6 Burst 3.4

7 Vocalization 669

8 Vocalization 1057.2

9 Vocalization 762

10 Vocalization 505

11 Noise 0.88

12 Swallowing

vibrations

18.6

13 Noise 3


Vo

cali

zati

on

s

1. 2. 3. 4. 5. 6.

2200 2250 2300 2350 2400

200

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300

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200

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300

350 5250 5300 5350 5400

150

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400

2800 2850 2900 2950

150

200

250

300

350

400

MSE=1999 MSE=988 MSE=1057,2 MSE=1050,1 MSE=2783,8 MSE=4583,8

7. 8. 9. 10. 11. 12.

4000 4050 4100 4150 4200

150

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400

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MSE=1439,1 MSE=3093,8 MSE=2219,4 MSE=1969 MSE=1983 MSE=1087,1

Sw

all

ow

ing

Vib

rati

on

s

13. 14. 15. 16. 17. 18.

1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500

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MSE=20 MSE=16 MSE=33.6 MSE=32 MSE=16 MSE=25

The illustrations of several zoomed time-frequency regions

and corresponding MSEs TEMPUS BioEMIS Edition, 2016 90

Telemedicine


Telemedicine O Humans tend to live longer

O Increased pressure on healthcare systems worldwide to provide higher quality health care

O Telemedicine promotes the use of multimedia services and systems for increasing the availability of care for patients

O Telemedicine provides a way for patients to be examined and treated, while the health care provider and the patient are at different physical locations.

O Telemedicine technologies - services to patients using signals that can be acquired over distances.

O Signal and image transmission, storage and processing are the major components of telemedicine.


Telemedicine

O Telenurcing

O Telepharmacy

O Telerehabilitation

O Teleradiology

O Telecardiology

O Telesurgery


Telenursing

O Provide home care to older adults and/or other patient groups, which preferred to stay in the comfort of their own homes.

O Generally, the patients welcome the use of multimedia systems to communicate with a nurse about their physical and psychological conditions.

O An analysis of telenursing in the case of metered dose inhalers in a geriatric population have shown that multimedia systems can provide most of services, and only small percentage require on-site visits.


Telepharmacy O Assumes providing pharmaceutical care to

patients and medication dispensing from distance.

O The telepharmacy services adhere to all official

regulations and services as traditional pharmacies,

including verification of drugs before dispensing

and patient counseling.

O Telepharmacy services maintain the same services

as the traditional pharmacies and provide

additional value-added features.

O Specifically, it has been shown that the utility of

telepharmacy services for education via video was

superior to education provided via written

instructions on an package insert.


Telerehabilitation O Telerehabilitation was established in 1997 when the

National Institute on Disability and Rehabilitation Research (U.S. Department of Education) ranked it as one of the top priorities for a newly established Rehabilitation Engineering Research Center (RERC).

O It covers diverse fields of investigations (e.g., intelligent therapeutic robots and other health gadgets)

O The main efforts are made to:

O provide telecommunication techniques to support rehabilitation services at a distance,

O then to provide technology for monitoring and evaluating the rehabilitation progress,

O and finally to provide technology for therapeutic intervention at a distance


Telerehabilitation

A typical telerehabilitation system


Patient

Doctor

Special telecardio

device

Telecardiology O Telecardiology involves merging technology with

cardiology in order to provide a patient with a proper medical care.

O Telecardiology is currently a well-developed medical discipline involving many different aspects of cardiology: O acute coronary syndromes,

O congestive heart failure,

O sudden cardiac arrest, arrhythmias).


Telecardiology

O It becomes an essential tool for cardiologists.

O Patient consultations with cardiologists via multimedia systems are becoming extremely common: a cardiologist receives many signals and images in real time to assess the patient condition

O Telecardiology has the two major aims: O to reduce the healthcare cost.

O to evaluate the efficiency of telecardiac tools (e.g., wireless ECG) at variable distances.

By accomplishing these two aims, telecardiology will enhance the psychological well-being of patients in addition to bridging the gap between rural areas and hospitals.


Telesurgery O Dissemination of new surgical skills and

techniques across the wide spectrum of practicing

surgeons is often difficult and time consuming,

especially because the practicing surgeons can be

located very far from large teaching centers.


O It allows:

O dissemination of expertise,

O widespread patient care,

O cost savings, and

O improved community care

O Telesurgery seems to be a powerful method for

performing minimally invasive surgery

O Operators using a telesurgery platform can complete

maneuvers with delays up to 500 ms.

O The emulated surgery in animals can be effectively

executed using either ground or satellite, while

keeping the satellite bandwidth above 5 Mb/s.

Telesurgery


Literature O J Gelijns of the IAEA publication (ISBN 978-92-0-131010-1):

Diagnostic Radiology Physics: A Handbook for Teachers and Students

O F. Zhang, G. Bi, and Y. Chen, “Tomography time-frequency

transform,” IEEE Transactions on Signal Processing, Vol. 50, No. 6,

June 2002, pp. 1289-1297

O M. Lustig, et al., "Sparse MRI: The application of compressed sensing

for rapid MR imaging," Magnetic resonance in medicine, vol. 58, pp.

1182-1195, 2007.

O Emmanuel J. Candès, Justin Romberg, Member, IEEE, and Terence

Tao, IEEE Transactions on Information Theory, vol. 52, no. 2, Feb.

2006

O S. Stankovic, I. Orovic, LJ. Stankovic, "Single-Iteration Algorithm for

Compressive Sensing Reconstruction," 21st Telecommunications

Forum TELFOR 2013, Novembar, 2013

O S. Stankovic, I. Orovic, LJ. Stankovic, "An Automated Signal

Reconstruction Method based on Analysis of Compressive Sensed

Signals in Noisy Environment," Signal Processing, vol. 104, Nov

2014, pp. 43 - 50, 2014 TEMPUS BioEMIS Edition, 2016 102

O S. Stankovic, I. Orovic, E. Sejdic, "Multimedia Signals and Systems:

Basic and Advance Algorithms for Signal Processing,"Springer-

Verlag, New York, 2015

O M. Brajovic, I. Orovic, M. Dakovic, S. Stankovic, "Gradient-based signal

reconstruction algorithm in the Hermite transform domain," Electronic

Letters, Accepted for publication, 2015

O S. Stankovic, LJ. Stankovic, I. Orovic, "Compressive sensing approach

in the Hermite transform domain," Mathematical Problems in

Engineering, Volume 2015 (2015), Article ID 286590, 9 pages,

http://dx.doi.org/10.1155/2015/286590 , 2015

Literature


O M. S. Nikjoo, C.M. Steele, E. Sejdić, T. Chau,” Automatic

discrimination between safe and unsafe swallowing using a

reputation-based classifier”, ResearchBioMedical Engineering,

2011

O C. M. Steele, E. Sejdic, T. Chau, “Noninvasive Detection of Thin-

Liquid Aspiration Using Dual-Axis Swallowing Accelerometry” DOI

10.1007/s00455-012-9418-9, 2012

O I. Orovic, S. Stankovic, T. Chau, C. M. Steele, E. Sejdic, "Time-

frequency analysis and Hermite projection method applied to

swallowing accelerometry signals," EURASIP Journal on

Advances in Signal Processing, Vol. 2010, Article ID 323125, 7

pages, 2010

O A. Szczepański, K. Saeed, A Mobile Device System for Early

Warning of ECG Anomalies, Sensors 2014, 14(6), 11031-11044;

doi:10.3390/s140611031

Literature


O Szczepański, A.; Saeed, K. Real-Time ECG signal feature

extraction for the proposition of abnormal beat detection—

Periodical signal extraction. Proceedings of the International

Conference on Biometrics and Kansei Engineering, Tokyo, Japan,

5–7 July 2013.

O

Literature


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